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Difference between revisions of "1960 AHSME Problems/Problem 9"

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[[Category:Introductory Algebra Problems]]

Latest revision as of 18:58, 17 May 2018

Problem

The fraction $\frac{a^2+b^2-c^2+2ab}{a^2+c^2-b^2+2ac}$ is (with suitable restrictions of the values of a, b, and c):

$\text{(A) irreducible}\qquad$

$\text{(B) reducible to negative 1}\qquad$

$\text{(C) reducible to a polynomial of three terms}\qquad$

$\text{(D) reducible to} \frac{a-b+c}{a+b-c}\qquad$

$\text{(E) reducible to} \frac{a+b-c}{a-b+c}$

Solution

Use the commutative property to get \[\frac{a^2+2ab+b^2-c^2}{a^2+2ac+c^2-b^2}\] Factor perfect square trinomials to get \[\frac{(a+b)^2-c^2}{(a+c)^2-b^2}\] Factor difference of squares to get \[\frac{(a+b+c)(a+b-c)}{(a+b+c)(a-b+c)}\] Cancel out like terms (with suitable restrictions of a, b, c) to get \[\frac{a+b-c}{a-b+c}\] The answer is $\boxed{\textbf{(E)}}$.

See Also

1960 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 8 		Followed by
Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions
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